When it first occurred to me that I wanted to reflect on last week’s teaching, it never actually occurred to me that I’d run into a series of interconnected skills that would essentially become the teaching theme for the week.
It started when I was asked to teach a student how to create equivalent ratios. She was learning to simplify fractions in preparation for learning to add, subtract, multiply, and divide fractions. I love teaching equivalent ratios, so it seemed like a fairly simple task. I should have known better. This poor student was never taught factoring or the rules for divisibility. Step one was to print off a copy of my rules of divisibility card for her. Step two was to teach her how to factor.
Part of the problem here was that earlier in the evening, she had been working on lowest common multiples, and as such had a very difficult time separating multiples from factors. This led to a definition session, because we weren’t going to get anywhere if she couldn’t keep the two straight.
- Factor: a number that can be multiplied by another number to equal the target number (at the elementary level, a factor will nearly always be smaller than the target number unless fractions and decimals are involved)
- Multiple: the result of multiplying the target number by another number (at the elementary level, a multiple will nearly always be larger than the target number unless fractions or decimals are involved)
We finally got the two concepts straight in her mind, and then I whipped out the rules of divisibility. She wasn’t too sure at first, but after using it on a couple of two-digit numbers, she was hooked.
Rules of divisibility
- 2: The ones digit is 2, 4, 6, 8, or 0.
- 3: Add the digits in the number together. If the resulting number is divisible by 3, then the original number is divisible by 3.
- 4: If the last two digits in the number are divisible by 4, then the original number is divisible by 4.
- 5: The ones digit is 5 or 0.
- 6: If the number is divisible by both 2 and 3, then it is divisible by 6.
- 9: Add the digits in the number together. If the resulting number is divisible by 9, then the original number is divisible by 9.
- 10: The ones digit is 0.
There is a recently published complicated method for determining divisibility by 7, but I choose not to confuse my students with that. Also, do not fall into the trap of thinking that any number divisible by 2 and 4 must be divisible by 8. If you find your self wondering about that one, just factor out the number 12. It’s divisible by both 2 and 4, but not 8.
Once I had her on the right path, I went on with my evening (only to discover another student in need of a rules of divisibility card) and my week. Then, I was confronted with a student who understood that fractions could be simplified, but really had no idea how to do it.
Super Becca to the rescue! I sat him down and taught him how to create equivalent ratios, watched him misapply his earlier learnings some more, retaught him and made him do more practice, and watched him stumble through to the right answers! I was very proud of him, and he can now add fractions with unlike denominators like a pro!
I think this is just one more sign that i really need to write those Pocket Becca math books the kids have been asking me to write…
